Various Applications of Hopf Bifurcations Matt Mulvehill, Kaleb Mitchell, Niko Lachman

Characteristics and Namesake  The Poincaré-Andronov-Hopf Bifurcation  A bifurcation is the point where the character of a solution to a differential equation changes  There are many types of bifurcation points  In class we learned about one of these types called a saddle-node bifurcation - Made up of the phase lines

Saddle vs. Hopf Bifurcations

Important Concepts  A limit cycle is a closed trajectory that solutions tend towards either as time goes to infinity or towards negative infinity.  “subcritical bifurcation”: limit cycle appears for negative values of the parameter; solutions tend away from the limit cycle  “supercritical bifurcation”: limit cycle appears for positive values of the parameter; solutions tend towards the limit cycle

Supercritical versus Subcritical Supercritical Bifurcation Subcritical Bifurcation  Re( λ)>0; sink at equilibrium value  Solutions are unstable closed orbits; amplitude of the orbit increases when parameter gets more negative  Re( λ)<0 ; source at equilibrium value  Solutions are stable closed orbits(periodic) ; amplitude of the orbit increases when parameter increases

Hopf Bifurcation Example  Non- autonomous  Non-linear  First order  Ordinary Differential Equation

Solving for Equilibrium Points  The equilibrium point for the system is (0,0) for any α (alpha)

Linearize the System  Use the Jacobian to linearize the system of differential equations  Evaluate the Jacobian at the equilibrium point (0,0)

Solve for the Eigenvalues of the System det = 0

Complex Eigenvalues What We know... Re(λ)<0, α<0 Stable Spiral Sink Re(λ)>0, α>0 Unstable Spiral Source Re(λ)=0, α=0 Unstable Center λ=α±iβ

What we don’t know…  What does the phase portrait look like when α changes?  How does this system of differential equations differ from ones that we have studied previously? …and how do we find out?

Supercritical or Subcritical?  Check α = 0 and plug points into the equations to determine if the bifurcation of the system is supercritical or subcritical. = -y + x 3 + xy 2 = y + yx 2 + y (0,10) dx/dt = -10 dy/dt = 1010 At α = 0, the phase portrait is a source due to the direction vector pointing away from the equilibrium point (0,0)

Subcritical Bifurcation Phase Portrait at α = 0

So what happened to the unstable center at Re(λ)=0? ITS STILL THERE! But only for small x and y values

All Phase Portraits for α, subcritical α < 0α = 0α > 0

Cool Picture

Brusselator Reaction Example

Jacobian

Eigenvalues

Eigenvalues cont. b - α^2 - 1 > 0 Source at the equilibrium point with level set b - α^2 - 1 = 0 Source-Level set ends/begins b - α^2 - 1 < 0 Stable sink

Lienard’s Equation

Linearizing the DE  Step 1: Find the equilibrium values; equate both differential equations to zero  Step 2: Linearize using the Jacobian

Analysis  The eigenvalues of the equations are:  Hopf Bifurcations have complex eigenvalues  If we choose a point far away from our equilibrium point we can learn whether the bifurcation is subcritical or supercritical  In this case there is a source at (0,0) which makes it a supercritical bifurcation